Lesson 1.4: Angles and Triangles

New Concept

This lesson introduces angles and triangles. You'll learn to classify them by their properties, measure angles in degrees, and use key rules, like the sum of angles in a triangle being $180\degree$, to solve for unknown values.

What’s next

Next, you'll apply these definitions with interactive examples and practice cards to master classifying and measuring angles and triangles.

Property

When two line segments meet at a point, they form an angle. The point where the two sides meet is called the vertex.

• If the sides are less open than a right angle, we call the angle acute.
• If the sides are more open than a right angle, we call the angle obtuse.
• If the two sides are totally open to form a straight line, we call the angle a straight angle.
• In a right angle, the sides are perpendicular.

Examples

• The tip of a slice of pie typically forms an acute angle.
• At 5:00, the hands of a clock form an obtuse angle.
• A perfectly straight pencil lying on a desk represents a straight angle.

Property

We use degrees to measure how open an angle is. The symbol for a degree is $\degree$. A right angle is $90\degree$, and a straight angle is $180\degree$. The degree measure of an acute angle is less than $90\degree$, and the degree measure of an obtuse angle is between $90\degree$ and $180\degree$. The measure of an angle does not depend on the orientation of the angle, or on how long the sides are. It only depends on how open the sides are.

Examples

• If two angles form a right angle ($90\degree$) and one angle is $35\degree$, the other angle must be $90\degree - 35\degree = 55\degree$.
• An angle that measures $125\degree$ is an obtuse angle because its measure is between $90\degree$ and $180\degree$.
• If two angles form a straight line ($180\degree$) and one is $100\degree$, the other must be $180\degree - 100\degree = 80\degree$.

Explanation

Degrees are like tiny measurement units for angles. A right angle has 90 of them, and a straight line has 180. The more open the angle, the more degrees it measures, regardless of how long its sides are.

Property

To use a protractor to measure an angle:

1. Place the center dot of the protractor at the vertex of the angle.
2. Line up the bottom line of the protractor along one of the sides of the angle.
3. Read the number where the other side of the angle meets the scale on the circular edge of the protractor.

Examples

• To measure an angle, place the protractor's center on the vertex and align one side with the $0\degree$ mark. If the other side points to the $45\degree$ mark, the angle is $45\degree$.
• When measuring an obtuse angle, if one side is at $0\degree$ and the other passes through $140\degree$, the angle's measure is $140\degree$.
• Remember that a protractor has two scales. Always start reading from the zero on the scale that aligns with the first side of your angle.

Explanation

A protractor is a half-circle ruler designed for angles. You align its center with the angle's corner and one side with the zero line. The other side will then point to the angle's measurement in degrees.

Property

Triangles can be classified by the size of their angles:

• A right triangle has one right angle ($90\degree$).
• In an acute triangle, all three angles are acute (less than $90\degree$).
• In an obtuse triangle, one of the angles is obtuse (greater than $90\degree$).

Examples

• A triangle with angles measuring $50\degree$, $60\degree$, and $70\degree$ is an acute triangle because all its angles are less than $90\degree$.
• A triangle with angles measuring $45\degree$, $45\degree$, and $90\degree$ is a right triangle.
• A triangle with angles measuring $30\degree$, $40\degree$, and $110\degree$ is an obtuse triangle.

Explanation

A triangle's name can come from its largest angle. If it contains a square corner ($90\degree$), it's a right triangle. If one angle is wide (obtuse), it's an obtuse triangle. If all angles are sharp (acute), it's an acute triangle.

Property

Triangles can be classified by the lengths of their sides:

• All three sides of an equilateral triangle are equal in length.
• In an isosceles triangle, two sides are equal in length.
• In a scalene triangle, all three sides have different lengths.

Examples

• A triangle with side lengths of 7 cm, 7 cm, and 7 cm is an equilateral triangle.
• A triangle with side lengths of 5 inches, 5 inches, and 8 inches is an isosceles triangle.
• A triangle with side lengths of 4 m, 6 m, and 9 m is a scalene triangle.

Explanation

A triangle's name can also come from its side lengths. 'Equilateral' means all sides are equal. 'Isosceles' means two sides are equal. 'Scalene' means no sides are equal—they all have different lengths.

Property

In any triangle, the angle opposite the longest side is the largest angle, and the angle opposite the shortest side is the smallest angle.

Inequality Symbols:

• $\gt$ means "is greater than"
• $\lt$ means "is less than"

Examples

• In a triangle with sides of length 6, 9, and 12, the largest angle is opposite the side of length 12, and the smallest angle is opposite the side of length 6.
• If a triangle has angles $A$, $B$, and $C$ opposite sides $a$, $b$, and $c$, and $A=100\degree$, $B=50\degree$, $C=30\degree$, then we know the side lengths are ordered as $a > b > c$.
• In an isosceles triangle, the two equal angles are opposite the two equal sides.

Property

• The sum of the angles in any triangle is $180\degree$.
• The base angles of an isosceles triangle are equal.
• All the angles of an equilateral triangle are equal.

Examples

• If a triangle has two angles measuring $40\degree$ and $80\degree$, the third angle must be $180\degree - (40\degree + 80\degree) = 60\degree$.
• An isosceles triangle has a vertex angle of $50\degree$. The two base angles are equal, so each one measures $(180\degree - 50\degree) \div 2 = 65\degree$.
• An equilateral triangle has three equal angles, so each angle must be $180\degree \div 3 = 60\degree$.

Explanation

If you tear off the three corners of any paper triangle and line them up, they always form a perfectly straight line, which measures $180\degree$. This universal rule helps you find any missing angle in a triangle.